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McLeish (1974) (see http://projecteuclid.org/download/pdf_1/euclid.aop/1176996608) proofs in his paper a CLT for martingale difference arrays, but I am a little bit confused by the assumptions. At page 1 he writes "...finite second moments are generally not assumed ...". But one of the assumption is that $\max_{i\leq k_n}|X_{ni}|$ is uniformly bounded in $L^2$ (i.e. $\sup_{n\geq 1}\mathbb{E}[|\max_{i\leq k_n}|X_{ni}||^2]<K<\infty$).

Also, in this book Theorem 5.2.3 (see https://books.google.at/books?id=Sqg-YPcpzLYC&pg=PA36&lpg=PA36&dq=flemming+counting+uniform+integrability&source=bl&ots=T1mhlvKVzu&sig=Su-fuZeO0ZcbLuqRXluvH_4B-CY&hl=de&sa=X&ved=0ahUKEwj3qsjrwZXVAhXLiRoKHXsWB2gQ6AEIMTAB#v=onepage&q=flemming%20counting%20uniform%20integrability&f=false) the author writes (one paragraph over the Theorem, which cannot be seen in the preview): "...we want a result without finite variances ... a result due to McLeish provides the basis for the necessary generalization"

Doesn't follow from the mentioned uniform boundedness that the variances are finite??

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I think the condition that bothers you is in Theorem 2.3, and is remedied by Corollary 2.7. I think (and I don't want to read this dense paper closely, so I'm not 100% sure) that in situations like Cor. 2.7 you can apply 2.3 to the truncated summands. In other words, 2.3 is a tool, and not the main result. The author could have been a little more helpful in providing a roadmap, or in phrasing his results better.

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  • $\begingroup$ Ok, but all books/papers are always referring to Theorem 2.3, so I think that should be the main result or at least the result which is most often used. This notes (math.arizona.edu/~sethuram/notes/wi_mart1.pdf) also referring to McLeish but use the assumption $\max_{1\leq i \leq k_n} |X_{ni}|\overset{L^1}{\longrightarrow} 0$ instead of the $L^2$-boundedness. The proof is basically the same. Does from that assumption also follow that the variances are finite? If not, the result would have been proven in basically the same way without finite variances. $\endgroup$ – RandomUser Jul 20 '17 at 7:18

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